The issue of when imperfect sampling frames can result in more efficient estimators of population totals than perfect frames is explored. Our analysis is based on an expression we call the difference score. We show how, when properly expanded it provides an illuminating basis for comparing a weighted estimator under an imperfect frame with that of a conventional estimator assuming the frame has been corrected. Specifically, the circumstances (i.e., population and frame characteristics) under which an imperfect frame results in estimates of population totals that are more precise than those from a perfect frame can in many cases be discerned by analytically examining the terms in the expansion of this difference score. In addition, a classification tree methodology was used to further explore circumstances under which imperfect frames result in more precise estimators. The results of this analytical study complement, strengthen, and in many cases explain those discovered in an earlier empirical investigation that lead to recommendations as to when to correct a frame or when to adjust for imperfection using a weighting methodology called the arc weight estimator.
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